The center of an ellipse is located at (3, 2) One focus is located at (6, 2) and its associated directrix is represented by the line x = 11 1/3 Given the standard form of the equation of an ellipse, what are the values of h, k, a, and b?

Step-by-step explanation:

This is really hard to try and explain over the internet here, but I'll do my best, assuming you have some experience with ellipses.

We are given the center of (3, 2), and the focus of (6, 2).  The focus is 3 units to the right of the center; the other focus is 3 units to the left of the center at (0, 2).  The focus has coordinates of (-ae, 2) and (ae, 2) from left to right and the distance between them then is 2ae.  There are 6 units between the 2 foci, so 2ae = 6.

The directrix is 8 1/3 units from the center to the right AND the left.  The directrices have distances of 2a/e between them.  There are 16 2/3 units between the 2 directrices, so

We can solve for the eccentricity of this ellipse given the fact that the eccentricity is the ratio of the distance between the foci to the distance between the directrices.  Therefore,

The 2's andd the a's cancel, leaving you with

That gives us then that

Going back to the identity for the distance between the foci, 2ae = 6, we fill in e and solve for a:

and

which gives us that a = 5

Now use the identity to solve for b:

and

and

b = 4

Because this is a horizontally stretched ellipse, it is of the form

we fill in as

Again, since you are expected to be able to solve for the equation of an ellipse at this advanced level, I am assuming that the equations I gave above as a means to solve for the different characteristics of an ellipse are familiar to you.  This is definitely NOT beginning conics!

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step-by-step explanation:

at least i think its

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step-by-step explanation:

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